Conway criterion

Prototile Octagon satisfying the Conway criterion. Sections AB and ED are shown in red, and the remaining segments are shown in color with a dot on the point of centrosymmetry.
A tessellation of the above prototile meeting the Conway criterion.

In the mathematical theory of tessellations, the Conway criterion, named for the English mathematician John Horton Conway, is a sufficient rule for when a prototile will tile the plane. It consists of the following requirements:[1] The tile must be a closed topological disk with six consecutive points A, B, C, D, E, and F on the boundary such that:

  • the boundary part from A to B is congruent to the boundary part from E to D by a translation T where T(A) = E and T(B) = D.
  • each of the boundary parts BC, CD, EF, and FA is centrosymmetric—that is, each one is congruent to itself when rotated by 180-degrees around its midpoint.
  • some of the six points may coincide but at least three of them must be distinct.[2]

Any prototile satisfying Conway's criterion admits a periodic tiling of the plane—and does so using only 180-degree rotations.[1] The Conway criterion is a sufficient condition to prove that a prototile tiles the plane but not a necessary one. There are tiles that fail the criterion and still tile the plane.[3]

Every Conway tile is foldable into either an isotetrahedron or a rectangle dihedron and conversely, every net of an isotetrahedron or rectangle dihedron is a Conway tile.[4][3]

  1. ^ a b Will It Tile? Try the Conway Criterion! by Doris Schattschneider Mathematics Magazine Vol. 53, No. 4 (Sep, 1980), pp. 224-233
  2. ^ Periodic Tiling: Polygons in General
  3. ^ a b Treks Into Intuitive Geometry: The World of Polygons and Polyhedra by Jin Akiyama and Kiyoko Matsunaga, Springer 2016, ISBN 9784431558415
  4. ^ Cite error: The named reference video was invoked but never defined (see the help page).